Unhooking the cable from the HD Receiver and connected that to the new Genie is simple enough. But what about coax cable for the Genie Mini? You'll need an RG6 coax cable from a SWM-compatible splitter to feed that new Mini. This assumes, of course, that you already have a SWM setup (which can be verified by pressing the "Dash " button on your remote. If it doesn't say "SWM Connected", then you really must get an installer out.
Generally, if you're not sure about this stuff, it's best to request an installer.